Scrambling in hyperbolic black holes: shock waves and pole-skipping

Abstract

Abstract We study the scrambling properties of (d + 1)-dimensional hyperbolic black holes. Using the eikonal approximation, we calculate out-of-time-order correlators (OTOCs) for a Rindler-AdS geometry with AdS radius ℓ, which is dual to a d-dimensional conformal field theory (CFT) in hyperbolic space with temperature T = 1/(2π ℓ). We find agreement between our results for OTOCs and previously reported CFT calculations. For more generic hyperbolic black holes, we compute the butterfly velocity in two different ways, namely: from shock waves and from a pole-skipping analysis, finding perfect agreement between the two methods. The butterfly velocity v B (T) nicely interpolates between the Rindler-AdS result v B T = 1 2 π ℓ = 1 d − 1 $$ {v}_B\left(T=\frac{1}{2\pi \ell}\right)=\frac{1}{d-1} $$ and the planar result v B T ≫ 1 ℓ = d 2 d − 1 $$ {v}_B\left(T\gg \frac{1}{\ell}\right)=\sqrt{\frac{d}{2\left(d-1\right)}} $$ .

Keywords

• Gauge-gravity correspondence
• Holography and condensed matter physics (AdS/CMT)